HyRES: Measuring Node Influence via Hyperspherical Representation Equilibrium Shift

Abstract

Identifying influential nodes in graph-structured data is a fundamental challenge. Traditional metrics ignore non-linear GNN semantics, while deep influence maximization methods require computationally expensive, simulation-based supervision. To bridge the gap between topological analysis and deep representation learning, we propose Hyperspherical Representation Equilibrium Shift (HyRES), a theoretically grounded, unsupervised framework that redefines node influence as a geometric displacement within the latent space. We conceptualize a trained GNN on a hyperspherical manifold as a physical system in thermodynamic equilibrium, stabilized by competing attractive and repulsive forces. We leverage Linear Response Theory to derive a closed-form approximation of the global representation shift caused by a node's removal. This mathematically decouples influence into the node's residual force and local structural stiffness, explaining why lower-degree bridges often exert greater global impact than redundant dense hubs. Empirically, HyRES enables efficient, near-linear time inference via Hessian-Vector Products, scaling seamlessly to large graphs. Extensive experiments demonstrate that HyRES outperforms state-of-the-art supervised baselines, exhibiting exceptional robustness on heterophilic graphs and under unknown diffusion dynamics. Code is available at: https://github.com/xianyt/HyRES.